Gradient-Based Search Mechanism for Optimizing Photolithograph Masks

ABSTRACT

A mechanism is provided for optimizing a photolithograph mask. A given target pattern is received. An initial fictitious mask is generated from the given target pattern and an initial value of α 2  is selected where the initial value of α 2  is used to determine a light intensity and a wafer image. The light intensity for each pixel in the initial fictitious mask and the wafer image for each pixel in the initial fictitious mask are then determined. A determination is then made as to whether a convergence has been achieved by comparing the wafer image generated from the fictitious mask to the given target pattern. Responsive to a convergence of the wafer image generated from the fictitious mask to the given target pattern, a final mask is generated to use to transfer an image to a wafer.

BACKGROUND

The present application relates generally to an improved data processingapparatus and method and more specifically to an apparatus and methodfor a gradient-based search mechanism for optimizing photolithographmasks.

Optical lithography is a crucial step in semiconductor manufacturing.The basic principle of optical lithography is quite similar to that ofchemistry-based photography. The images of the patterned photo-mask areprojected through the high-precision optical system onto the wafersurface, which is coated with a layer of light-sensitive chemicalcompound, e.g. photo-resist. The patterns are then formed on the wafersurface after complex chemical reactions and follow-on manufacturingsteps, such as wet or dry etching.

The resolution of the photo-lithography system (R) can be described bythe well-known Rayleigh's equation:

$R = \frac{k_{1}\lambda}{NA}$

in which λ is the wavelength of the light source, NA is the numericalaperture and k₁ is the factor describing the complexity of resolutionenhancement techniques. As the very-large-scale integration (VLSI)technology pushes further into nanometer region, the feasible wavelengthof the photo-lithographic system remains unchanged at 193 nm. Althoughthere is anticipation that extreme ultraviolet lithography (EUVL) withthe wavelength of 13 nm will replace traditional optical lithography,the availability of EUVL remains uncertain due to technical challengesand cost issues. On the other hand, the physical limit of drylithography of NA is 1.0. The recently introduced immersion lithographyhas bigger NA (1.2), but it is harder to achieve higher NA values. Thusit is commonly recognized that k₁ remains a cost effective knob toachieve finer resolution.

Due to the unavoidable diffraction, the optical lithography system islossy in the sense that only low frequency components of theelectromagnetic field can pass the optical system. As the gap betweenthe required feature size and lithography wavelength gets bigger, thefinal wafer images are quite different from the patterns on the mask. Inthe past few years, resolution enhancement techniques (RETs) have becomenecessary in order to achieve the required pattern density. Onewell-known RET is the optical proximity correction (OPC), in which themask patterns are intentionally “distorted” so that the desired imagecan be formed on the wafer. Other commonly used RETs are sub-wavelengthresolution assist features (SRAF) and phase-shift masks (PSM). Nowadays,considerable amount of computing power has to be dedicated to thesepost-layout processes (often referred as data prep). Large computerfarms have to spend weeks of central processing unit (CPU) time toperform data prep after a design is completed. However, all these RETmethods have one significant drawback: there is no guarantee theachieved results will be optimal. Furthermore, as the technology isfurther pushed, manufacturing variations (e.g., dose and focusvariations during the lithograph steps) have to be considered. However,it is quite challenging to systematically incorporate the processvariations into the traditional RETs.

On the other hand, this particular problem can be considered from adifferent angle. Instead of locally perturbing the pattern to compensatefor the loss, the mask pattern may be treated as the input to theoptical system, and the wafer image as the output. The task then becomeshow to “design” a mask so that the desired wafer image can be formed.This concept is often referred as “image design”, and was proposed over20 years ago. However, the problem itself is often ill-posed in thesense that more than one input can generate the same output. It can beshown that the size of the search space is well over 2^(1,000,000),which is even larger than the number of atoms in the observableuniverse. There were some early attempts to find a feasible solution tothis problem by using method such as simulated annealing, geneticalgorithms, and random pixel flipping. In recent years, the growingchallenges facing sub-wavelength lithography and the ever increasingcomplexity of traditional RETs have made this idea more attractive,which is often referred as “inverse lithography” or “computationallithography”. A gradient search based method was proposed to overcomethe excessive computational cost. However, the proposed method usednon-realistic assumptions regarding the optical system (incoherent andcoherent), while it is well-recognized that partially coherent modelsare the only acceptable model for the optical lithography system.Furthermore, the method was not demonstrated in a true industriallithography environment.

SUMMARY

In one illustrative embodiment, a method, in a data processing system,is provided for optimizing a photolithograph mask. The illustrativeembodiments receive a given target pattern. The illustrative embodimentsgenerate an initial fictitious mask and select an initial value of α₂.In the illustrative embodiments, the initial value of α₂ is used todetermine a light intensity and a wafer image. The illustrativeembodiments determine the light intensity for each pixel in the initialfictitious mask. The illustrative embodiments determine the wafer imagefor each pixel in the initial fictitious mask. The illustrativeembodiments determine if a convergence has been achieved by comparingthe wafer image generated from the fictitious mask to the given targetpattern. The illustrative embodiments generate a final mask to use totransfer an image to a wafer in response to a convergence of the waferimage generated from the fictitious mask to the given target pattern.

In other illustrative embodiments, a computer program product comprisinga computer useable or readable medium having a computer readable programis provided. The computer readable program, when executed on a computingdevice, causes the computing device to perform various ones, andcombinations of, the operations outlined above with regard to the methodillustrative embodiment.

In yet another illustrative embodiment, a system/apparatus is provided.The system/apparatus may comprise one or more processors and a memorycoupled to the one or more processors. The memory may compriseinstructions which, when executed by the one or more processors, causethe one or more processors to perform various ones, and combinations of,the operations outlined above with regard to the method illustrativeembodiment.

These and other features and advantages of the present invention will bedescribed in, or will become apparent to those of ordinary skill in theart in view of, the following detailed description of the exampleembodiments of the present invention.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

The invention, as well as a preferred mode of use and further objectivesand advantages thereof, will best be understood by reference to thefollowing detailed description of illustrative embodiments when read inconjunction with the accompanying drawings, wherein:

FIG. 1 depicts a pictorial representation of an example distributed dataprocessing system in which aspects of the illustrative embodiments maybe implemented;

FIG. 2 shows a block diagram of an example data processing system inwhich aspects of the illustrative embodiments may be implemented;

FIG. 3 illustrates a highly simplified schematic view of astate-of-the-art optical lithography process;

FIG. 4 depicts an optical system that uses a gradient-based searchmechanism for optimizing photolithograph masks for a given targetpattern in accordance with an illustrative embodiment;

FIG. 5 depicts an exemplary flow diagram of the operation performed by agradient-based search mechanism in accordance with an illustrativeembodiment;

FIG. 6 depicts an alternative exemplary flow diagram of the operationperformed by a gradient-based search mechanism in accordance with anillustrative embodiment; and

FIGS. 7A-7D depict one example of the optimizing photolithograph masksusing a gradient-based search mechanism in accordance with anillustrative embodiment.

DETAILED DESCRIPTION

The illustrative embodiments provide a mechanism for generaloptimization for inverse lithography. The illustrative embodimentapplies two transformations to convert a optimization function fromdiscrete domain to continuous domain so that a more efficient gradientsearching method can be used. Realistic models (partially coherent) areused for the optical system and an efficient gradient calculation methodusing fast Fourier transform (FFT) is derived. Furthermore, the proposedtransformation enables a gradual morph of the function from continuousto discrete, thus ensuring the optimality of the final solution.

As will be appreciated by one skilled in the art, the present inventionmay be embodied as a system, method, or computer program product.Accordingly, the present invention may take the form of an entirelyhardware embodiment, an entirely software embodiment (includingfirmware, resident software, micro-code, etc.) or an embodimentcombining software and hardware aspects that may all generally bereferred to herein as a “circuit,” “module” or “system.” Furthermore,the present invention may take the form of a computer program productembodied in any tangible medium of expression having computer usableprogram code embodied in the medium.

Any combination of one or more computer usable or computer readablemedium(s) may be utilized. The computer-usable or computer-readablemedium may be, for example, but not limited to, an electronic, magnetic,optical, electromagnetic, infrared, or semiconductor system, apparatus,device, or propagation medium. More specific examples (a non-exhaustivelist) of the computer-readable medium would include the following: anelectrical connection having one or more wires, a portable computerdiskette, a hard disk, a random access memory (RAM), a read-only memory(ROM), an erasable programmable read-only memory (EPROM or Flashmemory), an optical fiber, a portable compact disc read-only memory(CDROM), an optical storage device, a transmission media such as thosesupporting the Internet or an intranet, or a magnetic storage device.Note that the computer-usable or computer-readable medium could even bepaper or another suitable medium upon which the program is printed, asthe program can be electronically captured, via, for instance, opticalscanning of the paper or other medium, then compiled, interpreted, orotherwise processed in a suitable manner, if necessary, and then storedin a computer memory. In the context of this document, a computer-usableor computer-readable medium may be any medium that can contain, store,communicate, propagate, or transport the program for use by or inconnection with the instruction execution system, apparatus, or device.The computer-usable medium may include a propagated data signal with thecomputer-usable program code embodied therewith, either in baseband oras part of a carrier wave. The computer usable program code may betransmitted using any appropriate medium, including but not limited towireless, wireline, optical fiber cable, radio frequency (RF), etc.

Computer program code for carrying out operations of the presentinvention may be written in any combination of one or more programminglanguages, including an object oriented programming language such asJava™, Smalltalk™, C++ or the like and conventional proceduralprogramming languages, such as the “C” programming language or similarprogramming languages. The program code may execute entirely on theuser's computer, partly on the user's computer, as a stand-alonesoftware package, partly on the user's computer and partly on a remotecomputer or entirely on the remote computer or server. In the latterscenario, the remote computer may be connected to the user's computerthrough any type of network, including a local area network (LAN) or awide area network (WAN), or the connection may be made to an externalcomputer (for example, through the Internet using an Internet ServiceProvider). In addition, the program code may be embodied on a computerreadable storage medium on the server or the remote computer anddownloaded over a network to a computer readable storage medium of theremote computer or the users' computer for storage and/or execution.Moreover, any of the computing systems or data processing systems maystore the program code in a computer readable storage medium afterhaving downloaded the program code over a network from a remotecomputing system or data processing system.

The illustrative embodiments are described below with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems) and computer program products according to the illustrativeembodiments of the invention. It will be understood that each block ofthe flowchart illustrations and/or block diagrams, and combinations ofblocks in the flowchart illustrations and/or block diagrams, can beimplemented by computer program instructions. These computer programinstructions may be provided to a processor of a general purposecomputer, special purpose computer, or other programmable dataprocessing apparatus to produce a machine, such that the instructions,which execute via the processor of the computer or other programmabledata processing apparatus, create means for implementing thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

These computer program instructions may also be stored in acomputer-readable medium that can direct a computer or otherprogrammable data processing apparatus to function in a particularmanner, such that the instructions stored in the computer-readablemedium produce an article of manufacture including instruction meanswhich implement the function/act specified in the flowchart and/or blockdiagram block or blocks.

The computer program instructions may also be loaded onto a computer orother programmable data processing apparatus to cause a series ofoperational steps to be performed on the computer or other programmableapparatus to produce a computer implemented process such that theinstructions which execute on the computer or other programmableapparatus provide processes for implementing the functions/actsspecified in the flowchart and/or block diagram block or blocks.

The flowchart and block diagrams in the figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof code, which comprises one or more executable instructions forimplementing the specified logical function(s). It should also be notedthat, in some alternative implementations, the functions noted in theblock may occur out of the order noted in the figures. For example, twoblocks shown in succession may, in fact, be executed substantiallyconcurrently, or the blocks may sometimes be executed in the reverseorder, depending upon the functionality involved. It will also be notedthat each block of the block diagrams and/or flowchart illustration, andcombinations of blocks in the block diagrams and/or flowchartillustration, can be implemented by special purpose hardware-basedsystems that perform the specified functions or acts, or combinations ofspecial purpose hardware and computer instructions.

Thus, the illustrative embodiments may be utilized in many differenttypes of data processing environments including a distributed dataprocessing environment, a single data processing device, or the like. Inorder to provide a context for the description of the specific elementsand functionality of the illustrative embodiments, FIGS. 1 and 2 areprovided hereafter as example environments in which aspects of theillustrative embodiments may be implemented. While the descriptionfollowing FIGS. 1 and 2 will focus primarily on a single data processingdevice implementation of a gradient-based search mechanism foroptimizing photolithograph masks, this is only an example and is notintended to state or imply any limitation with regard to the features ofthe present invention. To the contrary, the illustrative embodiments areintended to include distributed data processing environments andembodiments in which photolithograph masks may be optimized using agradient-based search mechanism.

With reference now to the figures and in particular with reference toFIGS. 1-2, example diagrams of data processing environments are providedin which illustrative embodiments of the present invention may beimplemented. It should be appreciated that FIGS. 1-2 are only examplesand are not intended to assert or imply any limitation with regard tothe environments in which aspects or embodiments of the presentinvention may be implemented. Many modifications to the depictedenvironments may be made without departing from the spirit and scope ofthe present invention.

With reference now to the figures, FIG. 1 depicts a pictorialrepresentation of an example distributed data processing system in whichaspects of the illustrative embodiments may be implemented. Distributeddata processing system 100 may include a network of computers in whichaspects of the illustrative embodiments may be implemented. Thedistributed data processing system 100 contains at least one network102, which is the medium used to provide communication links betweenvarious devices and computers connected together within distributed dataprocessing system 100. The network 102 may include connections, such aswire, wireless communication links, or fiber optic cables.

In the depicted example, server 104 and server 106 are connected tonetwork 102 along with storage unit 108. In addition, clients 110, 112,and 114 are also connected to network 102. These clients 110, 112, and114 may be, for example, personal computers, network computers, or thelike. In the depicted example, server 104 provides data, such as bootfiles, operating system images, and applications to the clients 110,112, and 114. Clients 110, 112, and 114 are clients to server 104 in thedepicted example. Distributed data processing system 100 may includeadditional servers, clients, and other devices not shown.

In the depicted example, distributed data processing system 100 is theInternet with network 102 representing a worldwide collection ofnetworks and gateways that use the Transmission ControlProtocol/Internet Protocol (TCP/IP) suite of protocols to communicatewith one another. At the heart of the Internet is a backbone ofhigh-speed data communication lines between major nodes or hostcomputers, consisting of thousands of commercial, governmental,educational and other computer systems that route data and messages. Ofcourse, the distributed data processing system 100 may also beimplemented to include a number of different types of networks, such asfor example, an intranet, a local area network (LAN), a wide areanetwork (WAN), or the like. As stated above, FIG. 1 is intended as anexample, not as an architectural limitation for different embodiments ofthe present invention, and therefore, the particular elements shown inFIG. 1 should not be considered limiting with regard to the environmentsin which the illustrative embodiments of the present invention may beimplemented.

With reference now to FIG. 2, a block diagram of an example dataprocessing system is shown in which aspects of the illustrativeembodiments may be implemented. Data processing system 200 is an exampleof a computer, such as client 110 in FIG. 1, in which computer usablecode or instructions implementing the processes for illustrativeembodiments of the present invention may be located.

In the depicted example, data processing system 200 employs a hubarchitecture including north bridge and memory controller hub (NB/MCH)202 and south bridge and input/output (I/O) controller hub (SB/ICH) 204.Processing unit 206, main memory 208, and graphics processor 210 areconnected to NB/MCH 202. Graphics processor 210 may be connected toNB/MCH 202 through an accelerated graphics port (AGP).

In the depicted example, local area network (LAN) adapter 212 connectsto SB/ICH 204. Audio adapter 216, keyboard and mouse adapter 220, modem222, read only memory (ROM) 224, hard disk drive (HDD) 226, CD-ROM drive230, universal serial bus (USB) ports and other communication ports 232,and PCI/PCIe devices 234 connect to SB/ICH 204 through bus 238 and bus240. PCI/PCIe devices may include, for example, Ethernet adapters,add-in cards, and PC cards for notebook computers. PCI uses a card buscontroller, while PCIe does not. ROM 224 may be, for example, a flashbasic input/output system (BIOS).

HDD 226 and CD-ROM drive 230 connect to SB/ICH 204 through bus 240. HDD226 and CD-ROM drive 230 may use, for example, an integrated driveelectronics (IDE) or serial advanced technology attachment (SATA)interface. Super I/O (SIO) device 236 may be connected to SB/ICH 204.

An operating system runs on processing unit 206. The operating systemcoordinates and provides control of various components within the dataprocessing system 200 in FIG. 2. As a client, the operating system maybe a commercially available operating system such as Microsoft® Windows®XP (Microsoft and Windows are trademarks of Microsoft Corporation in theUnited States, other countries, or both). An object-oriented programmingsystem, such as the Java™ programming system, may run in conjunctionwith the operating system and provides calls to the operating systemfrom Java™ programs or applications executing on data processing system200 (Java is a trademark of Sun Microsystems, Inc. in the United States,other countries, or both).

As a server, data processing system 200 may be, for example, an IBM®eServer™ System p® computer system, running the Advanced InteractiveExecutive (AIX®) operating system or the LINUX® operating system(eServer, System p, and AIX are trademarks of International BusinessMachines Corporation in the United States, other countries, or bothwhile LINUX is a trademark of Linus Torvalds in the United States, othercountries, or both). Data processing system 200 may be a symmetricmultiprocessor (SMP) system including a plurality of processors inprocessing unit 206. Alternatively, a single processor system may beemployed.

Instructions for the operating system, the object-oriented programmingsystem, and applications or programs are located on storage devices,such as HDD 226, and may be loaded into main memory 208 for execution byprocessing unit 206. The processes for illustrative embodiments of thepresent invention may be performed by processing unit 206 using computerusable program code, which may be located in a memory such as, forexample, main memory 208, ROM 224, or in one or more peripheral devices226 and 230, for example.

A bus system, such as bus 238 or bus 240 as shown in FIG. 2, may becomprised of one or more buses. Of course, the bus system may beimplemented using any type of communication fabric or architecture thatprovides for a transfer of data between different components or devicesattached to the fabric or architecture. A communication unit, such asmodem 222 or network adapter 212 of FIG. 2, may include one or moredevices used to transmit and receive data. A memory may be, for example,main memory 208, ROM 224, or a cache such as found in NB/MCH 202 in FIG.2.

Those of ordinary skill in the art will appreciate that the hardware inFIGS. 1-2 may vary depending on the implementation. Other internalhardware or peripheral devices, such as flash memory, equivalentnon-volatile memory, or optical disk drives and the like, may be used inaddition to or in place of the hardware depicted in FIGS. 1-2. Also, theprocesses of the illustrative embodiments may be applied to amultiprocessor data processing system, other than the SMP systemmentioned previously, without departing from the spirit and scope of thepresent invention.

Moreover, the data processing system 200 may take the form of any of anumber of different data processing systems including client computingdevices, server computing devices, a tablet computer, laptop computer,telephone or other communication device, a personal digital assistant(PDA), or the like. In some illustrative examples, data processingsystem 200 may be a portable computing device which is configured withflash memory to provide non-volatile memory for storing operating systemfiles and/or user-generated data, for example. Essentially, dataprocessing system 200 may be any known or later developed dataprocessing system without architectural limitation.

FIG. 3 illustrates a highly simplified schematic view of astate-of-the-art optical lithography process. In optical system (Φ( ))302, light source 304 of wavelength λ projects through received mask (Θ)306 to achieve a spatially distributed intensity field (I) 308 on thewafer source. After chemical reactions of photo-resist (R( )) 310 on thewafer source, final wafer image (Z) 312 farms on the wafer source.

One problem of inverse lithography is to find a mask pattern (among anunknown number of solutions) so that the final wafer image is as closeto a predefined pattern as possible. Because the mask has a limitedresolution, it is usually spatially sampled. Thus, the illustrativeembodiment focus on traditional binary mask. Therefore, the value ateach mask pixel may be either 0 or 1. On the other hand, the wafer imageis also sampled at certain grid. Assuming the number of grids in x and ydirection are M and N, respectively. The exact values of M and N dependon the resolution and the size of the optical dimension (OD). For thestate-of-the-art sub-wavelength lithography, the values of M and N mayeasily exceed 1,000. Therefore, if treated as a discrete optimizationfunction, the size of the search space is over 2^(1,000,000). Thus, anybrute-force method quickly becomes intractable.

FIG. 4 depicts an optical system that uses a gradient-based searchmechanism for optimizing photolithograph masks for a given targetpattern in accordance with an illustrative embodiment. Gradient-basedsearch mechanism 402 in optical system 400 uses an objective function toquantify the goodness of input mask 404, which may be referred to as aninitial photolithographic mask. Gradient-based search mechanism 402 usesthe “sum of pixel difference” as the optimization objective functionF(θ), which is defined as:

${F(\theta)} = {\frac{1}{MN}{\sum\limits_{j = 1}^{MN}\left( {{\hat{z}}_{j} - z_{j}} \right)^{2}}}$

where {circumflex over (z)}_(j) is predefined as the desired pattern onthe wafer surface, where z_(j) represents the actual wafer imagegenerated by the given mask, where j is the pixel location within thegrid, where M is a number of grids in a x direction, and where N is anumber of grids in a y direction. Therefore, the mask optimizationfunction may be summarized as:

-   -   Finding θ_(j), j=1, 2, . . . , M N, such that F(θ) is minimized,    -   with θ_(j)ε {0, 1} and; z_(j)ε {0, 1}

Although the mask optimization function may be a discrete function, thediscrete function size is quite large, therefore, blindly applying adiscrete optimization package may not always be feasible. Thus,gradient-based search mechanism 402 uses transformations so that thediscrete function may be converted to continuous domain, so that moreefficient optimization methods may be applied.

In order to transform the function into a continuous function,gradient-based search mechanism 402 uses two transformations. In thefirst transformation, gradient-based search mechanism 402 transformslight intensity 406 on the surface of wafer 408 to binary wafer image.In other words, the binary wafer image is practically the photo-resistmodel itself. Gradient-based search mechanism 402 uses a function thatis a C₁ function (with continuous first-order derivatives) so that thegradient may be calculated, which is defined as follows:

${S_{1}\left( {{\theta;\alpha_{1}},t_{r}} \right)} = \frac{1}{1 + ^{{{- \alpha_{1}}\theta} + {\alpha_{1}t_{r}}}}$

where t_(r) is a predefined constant photo-resist (CTR) threshold value.The value of α₁ determines how “steep” the function is. The bigger valueα₁ is, the closer it is to the step function. However, regardless of thechoice of α₁, function S1(•) is always a C₁ function.

Gradient-based search mechanism 402 uses a second transformation totransform the function to a continuous domain, so that more efficientoptimization methods may be used. In other words, instead of optimizingthe true binary mask, gradient-based search mechanism 402 perform theoptimization on a fictitious continuous mask and uses the transformationto translate the fictitious mask into discrete domain. Since thetransformation is one-to-one, gradient-based search mechanism 402 easilyidentifies the desired binary mask. Gradient-based search mechanism 402uses the following logistic function to transform the function tocontinuous domain:

${S_{2}\left( {\theta;\alpha_{2}} \right)} = \frac{1}{1 + ^{{- \alpha_{2}}\theta}}$

It is clear that this transformation is also a C₁ function, thusgradient-based search mechanism 402 may easily calculate the gradient. Abenefit of using the second transformation function is thatgradient-based search mechanism 402 may dynamically “tune” the value ofα₂. Graph 450 illustrates that the smaller the value α₂ is, the “softer”the function is. The bigger the value of α₂, the “harder” (or closer todiscrete) the function becomes. As α₂ gets sufficiently large, thefunction becomes almost completely discrete. The value of α₂ may beconsidered sufficiently large when α₂ is either within a predeterminedrange of a predetermined value or equals the predetermined value.Therefore, gradient-based search mechanism 402 starts with a smaller α₂value and gradually increases the “stiffness” of the function byincreasing α₂. Using such a strategy helps to ensure the optimality ofthe final result.

In order to apply an efficient optimization method, gradient-basedsearch mechanism 402 first calculates a gradient. In order to calculatethe gradient, gradient-based search mechanism 402 calculates thederivative of the objective function F(•) with respect to pixel θ_(j).Note that the objective function is:

${F(\theta)} = {\frac{1}{MN}{\sum\limits_{j = 1}^{MN}\left( {{\hat{z}}_{j} - z_{j}} \right)^{2}}}$

where θ=[θ₁, θ₂, . . . , θ_(MN)] defines the mask. Gradient-based searchmechanism 402 then applies the following CTR equation:

${S_{1}\left( {{\theta;\alpha_{1}},t_{r}} \right)} = \frac{1}{1 + ^{{{- \alpha_{1}}\theta} + {\alpha_{1}t_{r}}}}$

so that the objective function becomes:

${F(\theta)} = {\frac{1}{MN}{\sum\limits_{j = 1}^{MN}\left( {{\hat{z}}_{j} - {S_{1}\left( I_{j} \right)}} \right)^{2}}}$

where I_(j) is the light intensity at pixel location j. Gradient-basedsearch mechanism 402 may then calculate the light intensity at locationj using the following formula:

$I_{j} = {\sum\limits_{k = 1}^{m}{\sigma_{k}{{\sum\limits_{i = 1}^{MN}{\varphi_{ijk}{S_{2}\left( \theta_{i} \right)}}}}^{2}}}$

by applying the second transformation of:

${S_{2}\left( {\theta;\alpha_{2}} \right)} = \frac{1}{1 + ^{{- \alpha_{2}}\theta}}$

Therefore, the support of θ_(i) is (−∞,+∞). In these equations, I_(j) isthe light intensity at pixel location j, θ is a specific pixel, m is aneigenvector of the kernel, M is a number of grids in a x direction, N isa number of grids in a y direction, S₂ is a logistic function, σ is theeigenvalue of the optical kernel function, φ is eigenvector of theoptical kernel function, k is the kernel index, and i is the grid index.

Gradient-based search mechanism 402 then calculates the gradient of thesum of pixel differences F with respect to the mask at location p usingthe following formula:

${\frac{\partial}{\partial\theta_{p}}{F(\theta)}} = {{- 2}{\sum\limits_{j = 1}^{MN}{\left( {{\hat{z}}_{j} - z_{j}} \right){S_{1}^{\prime}\left( I_{j} \right)}{\sum\limits_{k = 1}^{m}{\sigma_{k} \cdot \cdot \begin{bmatrix}{\frac{\partial}{\partial\theta_{p}}{\sum\limits_{i = 1}^{MN}{\varphi_{ijk}{{S_{2}\left( \theta_{i} \right)} \cdot {\sum\limits_{i = 1}^{MN}{\varphi_{ijk}^{*}{{S_{2}\left( \theta_{i} \right)}++}}}}}}} \\{\sum\limits_{i = 1}^{MN}{\varphi_{ijk}{{S_{2}\left( \theta_{i} \right)} \cdot \frac{\partial}{\partial\theta_{p}}}{\sum\limits_{i = 1}^{MN}{\varphi_{ijk}^{*}{S_{2}\left( \theta_{i} \right)}}}}}\end{bmatrix}}}}}}$

Note that in the above equation, gradient-based search mechanism 402substitutes any complex number c, |c|²=c·c*, where * denotes complexconjugate operator.

If gradient-based search mechanism 402 lets

${\frac{\partial}{\partial\theta_{p}}{\sum\limits_{i = 1}^{MN}{\varphi_{ijk}{S_{2}\left( \theta_{i} \right)}}}} = {\varphi_{pjk}{S_{2}^{\prime}\left( \theta_{p} \right)}}$and${\frac{\partial}{\partial\theta_{p}}{\sum\limits_{i = 1}^{MN}{\varphi_{ijk}^{*}{S_{2}\left( \theta_{i} \right)}}}} = {\varphi_{pjk}^{*}{S_{2}^{\prime}\left( \theta_{p} \right)}}$then:${\frac{\partial}{\partial\theta_{p}}{F(\theta)}} = {{- 2}{\sum\limits_{j = 1}^{MN}{\left( {{\hat{z}}_{j} - z_{j}} \right){S_{1}^{\prime}\left( I_{j} \right)}{\sum\limits_{k = 1}^{m}{\sigma_{k} \cdot \cdot \begin{bmatrix}{\varphi_{pjk}{S_{2}^{\prime}\left( \theta_{p} \right)}{\sum\limits_{i = 1}^{MN}{\varphi_{ijk}^{*}{{S_{2}\left( \theta_{i} \right)}++}}}} \\{\sum\limits_{i = 1}^{MN}{\varphi_{ijk}{S_{2}\left( \theta_{i} \right)}\varphi_{pjk}^{*}{S_{2}^{\prime}\left( \theta_{p} \right)}}}\end{bmatrix}}}}}}$${\frac{\partial}{\partial\theta_{p}}{F(\theta)}} = {{- 2}{S_{2}^{\prime}\left( \theta_{p} \right)}{\sum\limits_{j = 1}^{MN}{\left( {{\hat{z}}_{j} - z_{j}} \right){{S_{1}^{\prime}\left( I_{j} \right)} \cdot \cdot {\sum\limits_{k = 1}^{m}{\sigma_{k}\begin{bmatrix}{\varphi_{pjk}{\sum\limits_{i = 1}^{MN}{\varphi_{ijk}^{*}{{S_{2}\left( \theta_{i} \right)}++}}}} \\{\sum\limits_{i = 1}^{MN}{\varphi_{ijk}{S_{2}\left( \theta_{i} \right)}\varphi_{pjk}^{*}}}\end{bmatrix}}}}}}}$${\frac{\partial}{\partial\theta_{p}}{F(\theta)}} = {{- 2}{S_{2}^{\prime}\left( \theta_{p} \right)}{\sum\limits_{k = 1}^{m}{\sigma_{k}{\sum\limits_{j = 1}^{MN}{\left( {{\hat{z}}_{j} - z_{j}} \right){{S_{1}^{\prime}\left( I_{j} \right)} \cdot \cdot \begin{bmatrix}{\varphi_{pjk}{\sum\limits_{i = 1}^{MN}{\varphi_{ijk}^{*}{{S_{2}\left( \theta_{i} \right)}++}}}} \\{\sum\limits_{i = 1}^{MN}{\varphi_{ijk}{S_{2}\left( \theta_{i} \right)}\varphi_{pjk}^{*}}}\end{bmatrix}}}}}}}$

When gradient-based search mechanism 402 separates the contribution ofeach kernel k in the sum-of-coherent-system model, then:

${\frac{\partial}{\partial\theta_{p}}{F(\theta)}} = {{- 2}{\sum\limits_{k = 1}^{m}{\sigma_{k}{S_{2}^{\prime}\left( \theta_{p} \right)}{\sum\limits_{j = 1}^{MN}{\left( {{\hat{z}}_{j} - z_{j}} \right){{S_{1}^{\prime}\left( I_{j} \right)} \cdot \cdot \begin{bmatrix}{\varphi_{pjk}{\sum\limits_{i = 1}^{MN}{\varphi_{ijk}^{*}{{S_{2}\left( \theta_{i} \right)}++}}}} \\{\sum\limits_{i = 1}^{MN}{\varphi_{ijk}{S_{2}\left( \theta_{i} \right)}\varphi_{pjk}^{*}}}\end{bmatrix}}}}}}}$

Note that inner most summation actually defines the field at location jdue to the contribution of kernel k. To simplify the notation,gradient-based search mechanism 402 drops the kernel index k since thecontributions from different kernels may be summarized. Instead,gradient-based search mechanism 402 only expresses the contribution ofthe first kernel σ₁. Gradient-based search mechanism 402 also uses f_(j)to define the field at location j, such that:

$f_{j} = {\sum\limits_{i = 1}^{MN}{\varphi_{ij}{S_{2}\left( \theta_{i} \right)}}}$Thus:${\frac{\partial}{\partial\theta_{p}}{F(\theta)}} = {{- 2}\sigma_{1}{S_{2}^{\prime}\left( \theta_{p} \right)}{\sum\limits_{j = 1}^{MN}{\left( {{\hat{z}}_{j} - z_{j}} \right){{S_{1}^{\prime}\left( I_{j} \right)} \cdot \left\lbrack {{\varphi_{pj}f_{j}^{*}} + {\varphi_{pj}^{*}f_{j}}} \right\rbrack}}}}$

By defining an intermediate C_(i) as:

$C_{i} = {\sum\limits_{j}^{MN}{{\varphi_{ij} \cdot \left( {{\hat{z}}_{j} - z_{j}} \right)}{S_{1}^{\prime}\left( I_{j} \right)}f_{j}^{*}}}$

gradient-based search mechanism 402 identifies that:

φ_(pj) f _(j)*+φ_(pj) *f _(j)=2Real{φ_(pj) f _(j)*}=2Real{φ_(pj) *f_(j)}

In the above calculation, all variables except C_(i) are eitheravailable during the calculation of light intensity or are simplederivatives. By calculating the intermediate term C_(i) as shown above,gradient-based search mechanism 402 identifies the overall complexity ofcalculation C_(j) for each location j is O(MN). Obviously, to calculatethe term C_(j) at all locations, the complexity will become O(M²N²).

However, with the similarity of the inner loop between calculating theintermediate term C_(i) and the light intensity I_(j), gradient-basedsearch mechanism 402 can thus calculate the term C_(i) exactly the sameway as the calculation of light intensity at location i except thatgradient-based search mechanism 402 conjugates the kernel or the filed.Thus, gradient-based search mechanism 402 convolves to a morecomplicated product, instead of the simple mask. In other words, insteadof calculating C_(i) one by one, gradient-based search mechanism 402follows the same procedure of convolution and calculates all C_(i)s inone fast Fourier transform (FFT) operation.

Once C_(i) is known, gradient-based search mechanism 402 calculates anoverall derivative as follows:

${\frac{\partial}{\partial\theta_{p}}{F(\theta)}} = {{- 4}\sigma_{1}{S_{2}^{\prime}\left( \theta_{p} \right)}{Re}\; {al}\left\{ C_{p} \right\}}$

In these equations, θ is a specific pixel, σ is the eigenvalue of thekernel, S′₂ is a derivative logistic function, p is the mask location,and C_(p) is the continuous function at location p. The above derivationcan be easily extended to the partial coherent case when multiplekernels are involved.

${\frac{\partial}{\partial\theta_{p}}{F(\theta)}} = {{- 4}{\sum\limits_{k = 1}^{m}{\sigma_{k}{S_{2}^{\prime}\left( \theta_{p} \right)}{Re}\; {al}\left\{ {\sum\limits_{j}^{MN}{{\varphi_{ijk} \cdot \left( {{\hat{z}}_{j} - z_{j}} \right)}{S_{1}^{\prime}\left( I_{j} \right)}f_{jk}^{*}}} \right\}}}}$

Once gradient-based search mechanism 402 has calculated the gradient,then gradient-based search mechanism 402 uses a steepest decent methodfor optimization. Gradient-based search mechanism 402 calculates anoptimization at step w, i.e., θ_(w), where w is the index and theoptimization goes from step w, to w+1, to w+2, etc., then gradient-basedsearch mechanism 402 updates the mask with the following equation:

$\theta_{w + 1} = {\theta_{w} - {\gamma \frac{\partial{F\left( \theta_{w} \right)}}{\partial\theta}}}$

where θ is a specific pixel, where w is an increments step of θ, andwhere parameter γ is a damping parameter to control the step size.

Thus, since gradient-based search mechanism 402 may dynamically adjustthe parameter α₂ in the transformation in equation:

${S_{2}\left( {\theta;\mspace{11mu} \alpha_{2}} \right)} = \frac{1}{1 + ^{{- \alpha_{2}}\theta}}$

Then gradient-based search mechanism 402 may perform steepest descentoptimization in the inner loop and adjust α₂ in the outer loop, untilconvergence has been achieved by comparing the wafer image generated bythe fictitious mask to the given target. Note that eventuallygradient-based search mechanism 402 wants the α₂ value to besufficiently large so that the fictitious continuous function is veryclose to the original discrete optimization function. However,gradient-based search mechanism 402 may also combine two loops into oneand start with a small α₂ value and, as the steep descent method isreducing the overall objective function F, gradient-based searchmechanism 402 gradually increases α2.

FIG. 5 depicts an exemplary flow diagram of the operation performed by agradient-based search mechanism in accordance with an illustrativeembodiment. As the operation begins, the gradient-based search mechanisminitializes and receives a given target pattern (step 502). The gradientbased search mechanism then generates an initial fictitious mask (step504). The gradient-based search mechanism then selects an initial valueof α₂ (step 506). Using the chosen value of α₂, the gradient-basedsearch mechanism calculates a light intensity (step 508) using thefollowing equation:

$I_{j} = {\sum\limits_{k = 1}^{m}{\sigma_{k}{{\sum\limits_{i = 1}^{MN}{\varphi_{ijk}{S_{2}\left( \theta_{i} \right)}}}}^{2}}}$

Once the light intensity is calculated, the gradient-based searchmechanism calculates a wafer image (step 510) using the followingequation:

${S_{1}\left( {{\theta;\mspace{11mu} \alpha_{1}},t_{r}} \right)} = \frac{1}{1 + ^{{{- \alpha_{1}}\theta} + {\alpha_{1}t_{r}}}}$

The gradient-based search mechanism then calculates a difference betweenthe wafer image and the target pattern (step 512). Then gradient-basedsearch mechanism determines if convergence is achieved (step 514) usingthe following formula:

${F(\theta)} = {\frac{1}{MN}{\sum\limits_{f = 1}^{MN}\left( {{\hat{z}}_{j} - z_{j}} \right)^{2}}}$

That is, the convergence is measured by comparing the difference of agiven target pattern and the wafer image generated by the fictitiousmask. If at step 514 gradient-based search mechanism determines thatconvergence has failed to be achieved, the gradient-based searchmechanism calculates a gradient (step 516) using the following formula:

${\frac{\partial}{\partial\theta_{p}}{F(\theta)}} = {{- 4}\sigma_{1}{S_{2}^{\prime}\left( \theta_{p} \right)}{Re}\; {al}\left\{ C_{p} \right\}}$

Using a steepest decent, the gradient-based search mechanism uses thegradient to update the fictitious mask (step 518) using the followingformula:

$\theta_{w + 1} = {{\theta \;}_{w} - {\gamma \frac{\partial{F\left( \theta_{w} \right)}}{\partial\theta}}}$

With the operation retuning to step 508 thereafter. If at step 514convergence has been achieved, then the gradient-based search mechanismincreases the value of α₂ (step 520). The gradient based searchmechanism then determines if α₂ is sufficiently large (step 522). If atstep 522 α₂ is not sufficiently large, then the operation returns tostep 508. If at step 522 α₂ is sufficiently large, then thegradient-based search mechanism generates a final mask to use totransfer an image to a wafer (step 524), with the operation terminatingthereafter.

FIG. 6 depicts an alternative exemplary flow diagram of the operationperformed by a gradient-based search mechanism in accordance with anillustrative embodiment. As the operation begins, the gradient-basedsearch mechanism initializes and receives a given target pattern (step602). The gradient based search mechanism then generates an initialfictitious mask (step 604). The gradient-based search mechanism thenselects an initial value of α₂ (step 606). Using the chosen value of α₂,the gradient-based search mechanism calculates a light intensity (step608) using the following equation:

$I_{j} = {\sum\limits_{k = 1}^{m}{\sigma_{k}{{\sum\limits_{k = 1}^{MN}{\varphi_{ijk}{S_{2}\left( \theta_{i} \right)}}}}^{2}}}$

Once the light intensity is calculated, the gradient-based searchmechanism calculates a wafer image (step 610) using the followingequation:

${S_{1}\left( {{\theta;\alpha_{1}},t_{r}} \right)} = \frac{1}{{\, 1} + ^{{{- \alpha_{1}}\theta} + {\alpha_{1}t_{r}}}}$

The gradient-based search mechanism then calculates a difference betweenthe wafer image and the target pattern (step 612). Then thegradient-based search mechanism determines if α₂ is sufficiently largeand convergence is achieved (step 614) using the following formula:

${F(\theta)} = {\frac{1}{MN}{\sum\limits_{j = 1}^{MN}\left( {{\hat{z}}_{j} - z_{j}} \right)^{2}}}$

That is, the convergence is measured by comparing the difference of agiven target pattern and the wafer image generated by the fictitiousmask. If at step 614 the gradient-based search mechanism determines thatconvergence has failed to be achieved or α₂ is not sufficiently large,then the gradient-based search mechanism calculates a gradient (step616) using the following formula:

${\frac{\partial}{\partial\theta_{p}}{F(\theta)}} = {{- 4}\sigma_{1}{S_{2}^{\prime}\left( \theta_{p} \right)}{Re}\; {al}\left\{ C_{p} \right\}}$

Using a steepest decent, the gradient-based search mechanism uses thegradient to update the fictitious mask (step 618) using the followingformula:

$\theta_{w + 1} = {\theta_{w} - {\gamma \frac{\partial{F\left( \theta_{w} \right)}}{\partial\theta}}}$

Then the gradient-based search mechanism slightly increases the value ofα₂ (step 620) with the operation returning to step 608 thereafter. If atstep 614 α₂ is sufficiently large and convergence has been achieved,then the gradient-based search mechanism generates a final mask to useto transfer an image to a wafer (step 622), with the operationterminating thereafter.

FIGS. 7A-7D depict one example of the optimizing photolithograph masksusing a gradient-based search mechanism in accordance with anillustrative embodiment. FIG. 7A shows an intended target. After thegradient-based search mechanism completes the optimization stepsdescribed above, the final light intensity on the wafer surface is shownin FIG. 7B, which translates into the final binary wafer image as shownin FIG. 7C. Comparing the intended target, the wafer image in FIG. 7Cshows very good fidelity to the intended target in FIG. 7A. FIG. 7Dillustrates the mask generated by the gradient-based search mechanism.The thin vertical bar 702 on the right hand side of the FIG. 7D is tofacilitate the printing of the true vertical wiring. Vertical bar 702functions similar to sub-wavelength resolution assist features (SRAF).Gradient-based search mechanism automatically generates vertical bar 702during optimization to assist with mask generation; however, as shown,the vertical bar does not transfer to the wafer.

Thus, the illustrative embodiments provide mechanisms for generaloptimization for inverse lithography. The illustrative embodimentapplies two transformations to convert the optimization function fromdiscrete domain to continuous domain so that a more efficient gradientsearching method can be used. Realistic models (partially coherent) areused for the optical system and an efficient gradient calculation methodusing fast Fourier transform (FFT) is derived. Furthermore, the proposedtransformation enables a gradual morph of the function from continuousto discrete, thus ensuring the optimality of the final solution.

As noted above, it should be appreciated that the illustrativeembodiments may take the form of an entirely hardware embodiment, anentirely software embodiment or an embodiment containing both hardwareand software elements. In one example embodiment, the mechanisms of theillustrative embodiments are implemented in software or program code,which includes but is not limited to firmware, resident software,microcode, etc.

A data processing system suitable for storing and/or executing programcode will include at least one processor coupled directly or indirectlyto memory elements through a system bus. The memory elements can includelocal memory employed during actual execution of the program code, bulkstorage, and cache memories which provide temporary storage of at leastsome program code in order to reduce the number of times code must beretrieved from bulk storage during execution.

Input/output or I/O devices (including but not limited to keyboards,displays, pointing devices, etc.) can be coupled to the system eitherdirectly or through intervening I/O controllers. Network adapters mayalso be coupled to the system to enable the data processing system tobecome coupled to other data processing systems or remote printers orstorage devices through intervening private or public networks. Modems,cable modems and Ethernet cards are just a few of the currentlyavailable types of network adapters.

The description of the present invention has been presented for purposesof illustration and description, and is not intended to be exhaustive orlimited to the invention in the form disclosed. Many modifications andvariations will be apparent to those of ordinary skill in the art. Theembodiment was chosen and described in order to best explain theprinciples of the invention, the practical application, and to enableothers of ordinary skill in the art to understand the invention forvarious embodiments with various modifications as are suited to theparticular use contemplated.

1. A method, in a data processing system, for optimizing aphotolithograph mask, the method comprising: receiving, by the dataprocessing system, a given target pattern; generating, by the dataprocessing system, an initial fictitious mask; selecting, by the dataprocessing system, an initial value of α₂, wherein the initial value ofα₂ is used to determine a light intensity and a wafer image;determining, by the data processing system, the light intensity for eachpixel in the initial fictitious mask; determining, by the dataprocessing system, the wafer image for each pixel in the initialfictitious mask; determining, by the data processing system, if aconvergence has been achieved by comparing the wafer image generatedfrom the fictitious mask to the given target pattern; and responsive toa convergence of the wafer image generated from the fictitious mask tothe given target pattern, generating, by the data processing system, afinal mask to use to transfer an image to a wafer.
 2. The method ofclaim 1, further comprising: responsive to a failure of the wafer imagegenerated from the fictitious mask to converge with the given targetpattern, determining, by the data processing system, a gradient for eachpixel in the initial fictitious mask, wherein the gradient is determinedusing the following formula:${\frac{\partial}{\partial\theta_{p}}{F(\theta)}} = {{- 4}\sigma_{1}{S_{2}^{\prime}\left( \theta_{p} \right)}{{Re}{al}}\left\{ C_{p} \right\}}$wherein θ is a specific pixel, wherein σ is the eigenvalue of thekernel, wherein S′₂ is a derivative logistic function, wherein p is themask location, and wherein C_(p) is the continuous function at locationp; updating, by the data processing system, the initial fictitious maskusing the gradient thereby forming an updated fictitious mask; andrepeating, by the data processing system, the steps of determining thelight intensity for each pixel in the updated fictitious mask,determining the wafer image for each pixel in the updated fictitiousmask, and determining if the convergence has been achieved by comparingthe wafer image generated from the updated fictitious mask to the giventarget pattern.
 3. The method of claim 1, further comprising: responsiveto the wafer image generated from the fictitious mask converging withthe given target pattern, determining, by the data processing system,whether α₂ is either within a predetermined range of a predeterminedvalue or equal to the predetermined value; responsive to α₂ beinginsufficiently large, increasing, by the data processing system, thevalue of α₂; and repeating, by the data processing system, the steps ofdetermining the light intensity for each pixel in the initial fictitiousmask, determining the wafer image for each pixel in the initialfictitious mask, and determining if the convergence has been achieved bycomparing the wafer image generated from the initial fictitious mask tothe given target pattern.
 4. The method of claim 1, wherein the lightintensity is determined using the following equation:$I_{j} = {\sum\limits_{k = 1}^{m}{\sigma_{k}{{\sum\limits_{i = 1}^{MN}{\varphi_{ijk}{S_{2}\left( \theta_{i} \right)}}}}^{2}}}$wherein I_(j) is the light intensity at pixel location j, wherein θ is aspecific pixel, wherein m is an eigenvector of the kernel, wherein M isa number of grids in a x direction, wherein N is a number of grids in ay direction, wherein S₂ is a logistic function, wherein σ is theeigenvalue of the optical kernel function, wherein φ is eigenvector ofthe optical kernel function, wherein k is the kernel index, and whereini is the grid index.
 5. The method of claim 1, wherein the wafer imageis determined from light intensity using the following equation:${S_{1}\left( {{\theta;\alpha_{1}},t_{r}} \right)} = \frac{1}{1 + ^{{{- \alpha_{1}}\theta} + {\alpha_{1}t_{r}}}}$wherein S₁ is a logistic function and wherein θ is a specific pixel. 6.The method of claim 1, wherein the fictitious mask is determined usingthe following formula:$\theta_{w + 1} = {\theta_{w} - {\gamma \frac{\partial{F\left( \theta_{w} \right)}}{\partial\theta}}}$wherein θ is a specific pixel, wherein w is an increments step of θ, andwherein γ is a damping parameter to control the step size.
 7. The methodof claim 1, wherein determining if the convergence has been achieved bycomparing the wafer image generated from the fictitious mask to thegiven target pattern is determined using the following formula:${F(\theta)} = {\frac{1}{MN}{\sum\limits_{j = 1}^{MN}\left( {{\hat{z}}_{j} - z_{j}} \right)^{2}}}$wherein {circumflex over (z)}_(j) is predefined as the desired patternon the wafer surface, wherein z_(j) represents the actual wafer imagegenerated from the fictitious mask, wherein j is the pixel locationwithin the grid, wherein M is a number of grids in a x direction, andwherein N is a number of grids in a y direction.
 8. A computer programproduct comprising a computer recordable medium having a computerreadable program recorded thereon, wherein the computer readableprogram, when executed on a computing device, causes the computingdevice to: receive a given target pattern; generating an initialfictitious mask; select an initial value of α₂, wherein the initialvalue of α₂ is used to determined a light intensity and a wafer image;determine the light intensity for each pixel in the initial fictitiousmask; determine the wafer image for each pixel in the initial fictitiousmask; determine if a convergence has been achieved by comparing thewafer image generated from the fictitious mask to the given targetpattern; and responsive to a convergence of the wafer image generatedfrom the fictitious mask to the given target pattern, generate a finalmask to use to transfer an image to a wafer.
 9. The computer programproduct of claim 8, wherein the computer readable program further causesthe computing device to: responsive to a failure of the wafer imagegenerated from the fictitious mask to converge with the given targetpattern, determine a gradient for each pixel in the initial fictitiousmask, wherein the gradient is determined using the following formula:${\frac{\partial}{\partial\theta_{p}}{F(\theta)}} = {{- 4}\sigma_{1}{S_{2}^{\prime}\left( \theta_{p} \right)}{Real}\left\{ C_{p} \right\}}$wherein θ is a specific pixel, wherein σ is the eigenvalue of thekernel, wherein S′₂ is a derivative logistic function, wherein p is themask location, and wherein C_(p) is the continuous function at locationp; update the initial fictitious mask using the gradient thereby formingan updated fictitious mask; and repeat the steps of determining thelight intensity for each pixel in the updated fictitious mask,determining the wafer image for each pixel in the updated fictitiousmask, and determining if the convergence has been achieved by comparingthe wafer image generated from the updated fictitious mask to the giventarget pattern.
 10. The computer program product of claim 8, wherein thecomputer readable program further causes the computing device to:responsive to the wafer image generated from the fictitious maskconverging with the given target pattern, determine whether α₂ is eitherwithin a predetermined range of a predetermined value or equal to thepredetermined value; responsive to α₂ being insufficiently large,increase the value of α₂; and repeat the computer readable program stepsof determining the light intensity for each pixel in the initialfictitious mask, determining the wafer image for each pixel in theinitial fictitious mask, and determining if the convergence has beenachieved by comparing the wafer image generated from the initialfictitious mask to the given target pattern.
 11. The computer programproduct of claim 8, wherein the computer readable program to determinethe light intensity uses the following equation:$I_{j} = {\sum\limits_{k = 1}^{m}{\sigma_{k}{{\sum\limits_{i = 1}^{MN}{\varphi_{ijk}{S_{2}\left( \theta_{i} \right)}}}}^{2}}}$wherein I_(j) is the light intensity at pixel location j, wherein θ is aspecific pixel, wherein m is an eigenvector of the kernel, wherein M isa number of grids in a x direction, wherein N is a number of grids in ay direction, wherein S₂ is a logistic function, wherein σ is theeigenvalue of the optical kernel function, wherein φ is eigenvector ofthe optical kernel function, wherein k is the kernel index, and whereini is the grid index.
 12. The computer program product of claim 8,wherein the computer readable program to determine the wafer image fromlight intensity uses the following equation:${S_{1}\left( {{\theta;\alpha_{1}},t_{r}} \right)} = \frac{1}{1 + ^{{{- \alpha_{1}}\theta} + {\alpha_{1}t_{r}}}}$wherein S₁ is a logistic function and wherein θ is a specific pixel. 13.The computer program product of claim 8, wherein the computer readableprogram to determine the fictitious mask uses the following formula:$\theta_{w + 1} = {\theta_{w} - {\gamma \frac{\partial{F\left( \theta_{w} \right)}}{\partial\theta}}}$wherein θ is a specific pixel, wherein w is an increments step of θ, andwherein γ is a damping parameter to control the step size.
 14. Thecomputer program product of claim 8, wherein the computer readableprogram to determine if the convergence has been achieved by comparingthe wafer image generated from the fictitious mask to the given targetpattern uses the following formula:${F(\theta)} = {\frac{1}{MN}{\sum\limits_{j = 1}^{MN}\left( {{\hat{z}}_{j} - z_{j}} \right)^{2}}}$wherein {circumflex over (z)}_(j) is predefined as the desired patternon the wafer surface, wherein z_(j) represents the actual wafer imagegenerated from the fictitious mask, wherein j is the pixel locationwithin the grid, wherein M is a number of grids in a x direction, andwherein N is a number of grids in a y direction.
 15. An apparatus,comprising: a processor; and a memory coupled to the processor, whereinthe memory comprises instructions which, when executed by the processor,cause the processor to: receive an initial photolithograph mask and agiven target pattern; generate an initial fictitious mask; select aninitial value of α₂, wherein the initial value of α₂ is used todetermined a light intensity and a wafer image; determine the lightintensity for each pixel in the initial fictitious mask; determine thewafer image for each pixel in the initial fictitious mask; determine ifa convergence has been achieved by comparing the wafer image generatedfrom the fictitious mask to the given target pattern; and responsive toa convergence of the wafer image generated from the fictitious mask tothe given target pattern, generate a final mask to use to transfer animage to a wafer.
 16. The apparatus of claim 15, wherein theinstructions further cause the processor to: responsive to a failure ofthe wafer image generated from the fictitious mask to converge with thegiven target pattern, determine a gradient for each pixel in the initialfictitious mask, wherein the gradient is determined using the followingformula:${\frac{\partial}{\partial\theta_{p}}{F(\theta)}} = {{- 4}\sigma_{1}{S_{2}^{\prime}\left( \theta_{p} \right)}{Real}\left\{ C_{p} \right\}}$wherein θ is a specific pixel, wherein σ is the eigenvalue of thekernel, wherein S′₂ is a derivative logistic function, wherein p is themask location, and wherein C_(p) is the continuous function at locationp; update the initial fictitious mask using the gradient thereby formingan updated fictitious mask; and repeat the steps of determining thelight intensity for each pixel in the updated fictitious mask,determining the wafer image for each pixel in the updated fictitiousmask, and determining if the convergence has been achieved by comparingthe wafer image generated from the updated fictitious mask to the giventarget pattern.
 17. The apparatus of claim 15, wherein the instructionsfurther cause the processor to: responsive to the wafer image generatedfrom the fictitious mask converging with the given target pattern,determining whether α₂ is either within a predetermined range of apredetermined value or equal to the predetermined value; responsive toα₂ being insufficiently large, increase the value of α₂; and repeat thecomputer readable program steps of determining the light intensity foreach pixel in the initial fictitious mask, determining the wafer imagefor each pixel in the initial fictitious mask, and determining if theconvergence has been achieved by comparing the wafer image generatedfrom the initial fictitious mask to the given target pattern.
 18. Theapparatus of claim 15, wherein the instructions to determine the lightintensity uses the following equation:$I_{j} = {\sum\limits_{k = 1}^{m}{\sigma_{k}{{\sum\limits_{i = 1}^{MN}{\varphi_{ijk}{S_{2}\left( \theta_{i} \right)}}}}^{2}}}$wherein I_(j) is the light intensity at pixel location j, wherein θ is aspecific pixel, wherein m is an eigenvector of the kernel, wherein M isa number of grids in a x direction, wherein N is a number of grids in ay direction, wherein S₂ is a logistic function, wherein σ is theeigenvalue of the optical kernel function, wherein φ is eigenvector ofthe optical kernel function, wherein k is the kernel index, and whereini is the grid index.
 19. The apparatus of claim 15, wherein theinstructions to determine the wafer image from light intensity uses thefollowing equation:${S_{1}\left( {{\theta;\alpha_{1}},t_{r}} \right)} = \frac{1}{1 + ^{{{- \alpha_{1}}\theta} + {\alpha_{1}t_{r}}}}$wherein S₁ is a logistic function and wherein θ is a specific pixel. 20.The apparatus of claim 15, wherein the instructions to determine thefictitious mask uses the following formula:$\theta_{w + 1} = {\theta_{w} - {\gamma \frac{\partial{F\left( \theta_{w} \right)}}{\partial\theta}}}$wherein θ is a specific pixel, wherein w is an increments step of θ, andwherein γ is a damping parameter to control the step size and whereinthe instructions to determine if the convergence has been achieved bycomparing the wafer image generated from the fictitious mask to thegiven target pattern uses the following formula:${F(\theta)} = {\frac{1}{MN}{\sum\limits_{j = 1}^{MN}\left( {{\hat{z}}_{j} - z_{j}} \right)^{2}}}$wherein {circumflex over (z)}_(j) is predefined as the desired patternon the wafer surface, wherein z_(j) represents the actual wafer imagegenerated from the fictitious mask, wherein j is the pixel locationwithin the grid, wherein M is a number of grids in a x direction, andwherein N is a number of grids in a y direction.